A199478: Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1,i)+binomial(n-i,i)), n >= 1, 0 <= i <= (n+1)/2.
A199478 ↗ nonnegative coefficients
Proved for all n all roots real, in (−∞, 0)coefficients of one signHadamard kernelinterlacing (weak)
Recurrence from A199478
\(P_{n}(x) = \left(1\right)\,P_{n-1}(x) + \left(2\,x\right)\,P_{n-2}(x)\)
\(P_{0}(x) = 1 + x\)
\(P_{1}(x) = 1 + 3\,x\)
Rows ?
| n | x0 | x1 | x2 | x3 | x4 | Pn(1) |
|---|---|---|---|---|---|---|
| 0 | 1 | 1 | 2 | |||
| 1 | 1 | 3 | 4 | |||
| 2 | 1 | 5 | 2 | 8 | ||
| 3 | 1 | 7 | 8 | 16 | ||
| 4 | 1 | 9 | 18 | 4 | 32 | |
| 5 | 1 | 11 | 32 | 20 | 64 | |
| 6 | 1 | 13 | 50 | 56 | 8 | 128 |
| 7 | 1 | 15 | 72 | 120 | 48 | 256 |
All 15 rows b-files: A199478 ?
Compute more
?
Technical details
- Node id
n-ff9680a2cfd23e93· revision 0 · JSON- Definition source
oeis_catalog:A199478- Fingerprint
f6-1baf5f9ce9075389
Observed on the stored rows ?
- Degrees: 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8 (half affine: slope 1, intercept 2)
- Palindromic rows: no
- γ-positive (gamma_positive): no
- Nonnegative coefficients (
nonn): yes - Only rational roots (
rr_by_factors: every nonzero row is a product of linear factors over ℚ): no
Matches ?
Row sums ? A050732, A090129, A155559, A011782, A034008, A118655, A131577, A000079
Submission text ?
Atlas rows n = 0, …, ascending powers of x
%e ExampleTriangle begins: 0: 1, 1; 1: 1, 3; 2: 1, 5, 2; 3: 1, 7, 8; 4: 1, 9, 18, 4; 5: 1, 11, 32, 20; 6: 1, 13, 50, 56, 8; 7: 1, 15, 72, 120, 48; 8: 1, 17, 98, 220, 160, 16; 9: 1, 19, 128, 364, 400, 112;
%F FormulaThe row polynomials satisfy P(n,x) = P(n-1,x) + 2*x*P(n-2,x) for n >= 2, with P(0,x) = 1 + x, P(1,x) = 1 + 3*x.
%t MathematicaP[0, x_] := 1 + x;
P[1, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 2*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten%o PARI/GPP(N) = {my(v = vector(N + 1)); v[1] = 1 + x; v[2] = 1 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = v[i - 1] + 2*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)polytool c48629374597
A199478 triangle, offset 1, from P0
%e ExampleTriangle begins: 1: 1, 1; 2: 1, 3; 3: 1, 5, 2; 4: 1, 7, 8; 5: 1, 9, 18, 4; 6: 1, 11, 32, 20; 7: 1, 13, 50, 56, 8; 8: 1, 15, 72, 120, 48; 9: 1, 17, 98, 220, 160, 16; 10: 1, 19, 128, 364, 400, 112;
%F FormulaThe row polynomials satisfy P(n,x) = P(n-1,x) + 2*x*P(n-2,x) for n >= 3, with P(1,x) = 1 + x, P(2,x) = 1 + 3*x.
%t MathematicaP[1, x_] := 1 + x;
P[2, x_] := 1 + 3*x;
P[n_, x_] := P[n, x] = Expand[P[n - 1, x] + 2*x*P[n - 2, x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten%o PARI/GPP(N) = {my(v = vector(N)); v[1] = 1 + x; v[2] = 1 + 3*x; for(n = 3, N, my(i = n); v[i] = v[i - 1] + 2*x*v[i - 2]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)polytool c48629374597
Properties
- Real-rooted? Every row P_n, n ≥ 0, is zero or real-rooted with all roots in (−∞, 0)
-
Coefficients of one sign?
Every nonzero row, n ≥ 0: the coefficients of P_n have one sign
derived from Liu–Wang theorem for three-term recurrences
-
Hadamard kernel (Malo)?
Rows P_n, n ≥ 0, are Hadamard kernels (preserve real-rootedness)
derived from Liu–Wang theorem for three-term recurrences
- Interlacing of consecutive rows? Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m ≥ 1
Proof chains ? JSON
Used in 10
- basis_map_A013609 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomia… Image under a basis map u^k -> P_k (A013609)
- basis_map_A013611 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomia… Image under a basis map u^k -> P_k (A013611)
- basis_map_A008277 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomia… Image under a basis map u^k -> P_k (A008277)
- basis_map_A013610 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomia… Image under a basis map u^k -> P_k (A013610)
- basis_map_A007318 of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomia… Image under a basis map u^k -> P_k (A007318)
- shift_x_plus_one of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial… Image under a real-rootedness preserver (affine, shift 0)
- derivative of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n-i+1… Image under a real-rootedness preserver (derivative polynomial, shift 0)
- theta_plus_one of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binomial(n… Image under a real-rootedness preserver (euler polynomial, shift 0)
2 more
- theta_plus_one_squared of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(bi… Image under a real-rootedness preserver (euler polynomial, shift 0)
- one_plus_derivative of Irregular triangle read by rows: T(n,i) = 2^(i-1)*(binom… Image under a real-rootedness preserver (derivative polynomial, shift 0)
References
- [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003